Central Potentials and Hydrogenic Systems

Central potentials are the canonical meeting point of three-dimensional wave mechanics, rotational symmetry, radial differential equations, and atomic structure. Every scalar potential of the form $V(\mathbf r)=V(r)$ shares the same angular organization, while its radial function determines the actual spectrum. The Coulomb potential is the most important exactly solvable example, but it is also unusually symmetric and should not be mistaken for a generic central potential.

This chapter first develops the reusable central-potential framework and its half-line boundary conditions. It then specializes to the point-Coulomb interaction, the nonrelativistic hydrogen atom, hydrogenic ions, radial functions, orbitals, the special $n^2$ degeneracy, and the positive-energy continuum.

Scope and model hierarchy

The chapter owns the spinless nonrelativistic Schrödinger treatment of one relative coordinate in a scalar central potential. For the Coulomb system, the baseline model uses point charges, reduced mass, and no external fields.

It does not silently include precision atomic physics. Fine structure, the Dirac equation, spin–orbit coupling, hyperfine structure, the Lamb shift, finite nuclear size, Zeeman and Stark effects, radiative transitions, and many-electron correlations belong in their dedicated symmetry, relativistic, field-theory, or atomic-physics treatments. Those effects refine or replace a clearly stated Coulomb baseline.

The angular division of labor is equally important. This chapter uses $\ell$, $m$, and spherical harmonics to solve wave-mechanics models. Symmetry, Angular Momentum, and Spin owns the operator algebra of rotations, angular-momentum multiplets, and hidden-symmetry interpretation.

Generic central potentials

A central-potential Hamiltonian for one relative coordinate is

$$H = \frac{\mathbf p^2}{2\mu} +V(r), \qquad r=\lvert\mathbf r\rvert.$$

Here $\mu$ is the particle mass for a one-body model or the reduced mass after a two-body center-of-mass separation. Because rotations preserve $r$,

$$[H,L_i]=0, \qquad [H,\mathbf L^2]=0.$$

Energy eigenfunctions can therefore be chosen in separated form:

$$\psi_{\nu\ell m}(r,\theta,\phi) = R_{\nu\ell}(r)Y_\ell^m(\theta,\phi).$$

The generic radial label $\nu$ may be a radial node number, a discrete energy label, or a continuum wavenumber. The angular labels obey

$$\ell=0,1,2,\ldots, \qquad m=-\ell,-\ell+1,\ldots,\ell.$$

The spherical harmonics satisfy

$$\mathbf L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad L_zY_\ell^m = \hbar mY_\ell^m.$$

For fixed radial data and $\ell$, the $2\ell+1$ values of $m$ have the same energy. This magnetic degeneracy follows from rotations. Degeneracy between different values of $\ell$ is not generic because changing $\ell$ changes the radial equation.

Central Potentials develops the physical meaning of this structure, the distinction among radial labels, and the bound-versus-scattering possibilities.

The radial half-line problem

The radial factor obeys

$$-\frac{\hbar^2}{2\mu} \frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{dR_\ell}{dr} \right) + \left[ V(r) + \frac{\hbar^2\ell(\ell+1)}{2\mu r^2} \right]R_\ell = ER_\ell.$$

Defining

$$u_\ell(r)=rR_\ell(r)$$

removes the first-derivative structure:

$$-\frac{\hbar^2}{2\mu} \frac{d^2u_\ell}{dr^2} + V_{\ell,\mathrm{eff}}(r)u_\ell = Eu_\ell,$$

where

$$V_{\ell,\mathrm{eff}}(r) = V(r) + \frac{\hbar^2\ell(\ell+1)}{2\mu r^2}.$$

This is a half-line equation on $r\gt0$, not a full-line one-dimensional model. The centrifugal term is angular kinetic energy written in radial form. It vanishes for $\ell=0$ and suppresses short-distance probability for $\ell\gt0$.

The two radial functions use different measures:

$$\int_0^\infty \lvert R_\ell(r)\rvert^2r^2,dr = \int_0^\infty \lvert u_\ell(r)\rvert^2,dr.$$

Radial Schrödinger Equation is the canonical derivation. Effective Radial Potential uses the fixed-$\ell$ equation to interpret centrifugal barriers, turning points, radial nodes, bound-state wells, and low-energy partial-wave suppression.

Radial boundary conditions

The origin is part of the operator domain. For a potential finite at the origin or milder than $1/r^2$, including the Coulomb potential, the two local powers are

$$R_\ell(r)\sim r^\ell \qquad\text{or}\qquad R_\ell(r)\sim r^{-\ell-1}.$$

The standard regular branch is

$$R_\ell(r)\sim r^\ell, \qquad u_\ell(r)\sim r^{\ell+1}, \qquad u_\ell(0)=0.$$

For a bound state, $u_\ell$ must also be square-integrable and decay at infinity. For a continuum state, decay is replaced by an incoming, outgoing, or standing-wave asymptotic convention. Finite jumps, delta shells, hard boundaries, and singular potentials each impose their own matching or domain conditions.

Potentials behaving as $-g/r^2$ require special care because they compete directly with the centrifugal term. The usual regular-potential rule cannot be imported without checking self-adjointness and short-distance physics.

Boundary Conditions for Radial Wavefunctions collects the endpoint, normalization, matching, current, and singular-potential checklist.

Reading an effective radial potential

For fixed $\ell$, radii satisfying

$$E=V_{\ell,\mathrm{eff}}(r)$$

are semiclassical turning points. Regions with $E\gt V_{\ell,\mathrm{eff}}$ are classically allowed in the radial sense, while regions with $E\lt V_{\ell,\mathrm{eff}}$ are classically forbidden. The exact quantum solution remains smooth through an ordinary turning point.

Within one regular $\ell$ sector, bound states are ordered by the number of interior zeros of $u_\ell$:

$$n_r=0,1,2,\ldots .$$

For a generic central potential the spectrum has the form

$$E=E_{n_r\ell}.$$

The node number $n_r$ counts radial nodes only. Angular nodal structure comes from the spherical harmonic. Any stronger degeneracy relating different $\ell$ sectors requires special structure beyond ordinary rotational symmetry.

The Coulomb interaction

For charges $q_1$ and $q_2$,

$$V(r) = \frac{q_1q_2}{4\pi\epsilon_0r}.$$

Like charges repel and opposite charges attract. An electron of charge $-e$ bound to a point nucleus of charge $+Ze$ has

$$V(r) = -\frac{Ze^2}{4\pi\epsilon_0r} = -\frac{\kappa_Z}{r}, \qquad \kappa_Z = \frac{Ze^2}{4\pi\epsilon_0}.$$

After center-of-mass separation, the reduced mass is

$$\mu = \frac{m_eM_N}{m_e+M_N}.$$

The natural length and energy scales are

$$a_Z = \frac{\hbar^2}{\mu\kappa_Z} = \frac{4\pi\epsilon_0\hbar^2}{\mu Ze^2},$$

and

$$E_Z = \frac{\mu\kappa_Z^2}{2\hbar^2} = \frac{\mu Z^2e^4} {2(4\pi\epsilon_0)^2\hbar^2}.$$

Thus lengths scale as $(\mu Z)^{-1}$ and binding energies as $\mu Z^2$. In dimensionless units $\rho=r/a_Z$, every attractive point-Coulomb one-electron system has the same equation; $Z$ and $\mu$ re-enter only when converting back to physical units.

Coulomb Potential fixes the charge signs, scales, reduced-mass convention, and bound-continuum split before the exact hydrogen solution is used.

The nonrelativistic hydrogen solution

For hydrogen, $Z=1$ and $M_N=m_p$. The relative Hamiltonian is

$$H = -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{4\pi\epsilon_0r}.$$

The spinless bound states are

$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi),$$

with

$$\begin{aligned} n&=1,2,3,\ldots,\\ \ell&=0,1,\ldots,n-1,\\ m&=-\ell,-\ell+1,\ldots,\ell. \end{aligned}$$

Their energies are

$$E_n = -\frac{\mu e^4} {2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2} = -\frac{E_1^{\mathrm{scale}}}{n^2}.$$

For ordinary hydrogen this is approximately $-13.6\,\mathrm{eV}/n^2$, with reduced mass included for accurate values. The bound levels accumulate at the ionization threshold $E=0$ from below.

The radial function has the structural form

$$R_{n\ell}(r) \propto e^{-\rho/2} \rho^\ell L_{n-\ell-1}^{(2\ell+1)}(\rho), \qquad \rho=\frac{2r}{na_0}.$$

Exponential decay controls the tail, $\rho^\ell$ enforces regularity, and the generalized Laguerre polynomial supplies

$$n_r=n-\ell-1$$

radial nodes. Hydrogen Atom gives the canonical bound-state model. Radial Wavefunctions is the formula and expectation-value reference.

Hydrogenic ions and scaling

Hydrogenic ions are one-electron systems such as $\mathrm{He}^+$ and $\mathrm{Li}^{2+}$, not neutral many-electron atoms. Replacing $Z$ and the nuclear mass leaves the dimensionless wavefunctions unchanged:

$$R_{n\ell}^{(Z)}(r) = a_Z^{-3/2} \mathcal R_{n\ell} \left( \frac{r}{a_Z} \right).$$

Ignoring the small reduced-mass differences for a moment,

$$a_Z\approx\frac{a_0}{Z}, \qquad E_n^{(Z)}\approx Z^2E_n^{(1)}.$$

At fixed quantum numbers, larger $Z$ makes the state smaller and more tightly bound. Relativistic and finite-nuclear-size corrections become increasingly important as $Z\alpha$ grows, so the Schrödinger scaling is a baseline rather than a precision theory at high $Z$.

Hydrogenic Ions develops length, energy, wavefunction, and transition-frequency scaling with the reduced mass kept explicit.

Radial probability and orbitals

The radial factor is not a radial probability density. With unit-normalized angular functions,

$$dP = r^2\lvert R_{n\ell}(r)\rvert^2,dr = \lvert u_{n\ell}(r)\rvert^2,dr.$$

For the hydrogenic ground state,

$$R_{10}(r) = 2a_0^{-3/2}e^{-r/a_0},$$

so the full wavefunction is largest at the origin while the radial probability is maximal at $r=a_0$. These are different statements because a spherical shell has volume proportional to $r^2dr$.

An orbital is a one-electron spatial wavefunction, not a trajectory or a hard-edged cloud. The letters $s,p,d,f,\ldots$ encode

$$\ell=0,1,2,3,\ldots .$$

Complex $Y_\ell^m$ orbitals diagonalize $L_z$. Familiar real orbitals are linear combinations inside a degenerate $m$ subspace. Shading usually displays wavefunction sign or phase, while probability is $\lvert\psi\rvert^2$.

Atomic Orbitals is the canonical page for spectroscopic letters, real and complex bases, radial and angular nodes, and visualization cautions. Many-electron orbitals are approximate one-electron objects inside a many-body theory and should not be identified with exact hydrogen eigenstates.

Why the hydrogen spectrum is unusually degenerate

For fixed $n$, the ideal spinless Coulomb Hamiltonian has

$$g_n = \sum_{\ell=0}^{n-1}(2\ell+1) = n^2$$

spatial bound states. The degeneracy has two distinct sources:

Degeneracy	States related	Explanation
Rotational	Different $m$ at fixed $n,\ell$	Ordinary central-potential symmetry
Coulomb-specific	Different $\ell$ at fixed $n$	Hidden inverse-radius structure

For a generic central potential, changing $\ell$ changes the centrifugal term and changes the radial energy. In the Coulomb problem,

$$n=n_r+\ell+1,$$

and the energy depends only on this combination. The quantum Laplace–Runge–Lenz structure enlarges the bound-state symmetry, often described by $\mathrm{SO}(4)$, and relates different $\ell$ sectors.

If electron spin is included but every spin-dependent interaction is artificially absent, the count doubles to $2n^2$. Real relativistic, radiative, hyperfine, recoil, finite-size, and external-field effects split the ideal levels in different patterns.

Degeneracy of the Hydrogen Atom is the canonical explanation of the count, the rotational part, the hidden-symmetry part, and the hierarchy of degeneracy-lifting effects.

Bound states and continuum completion

The attractive Coulomb spectrum is not only the discrete hydrogen ladder:

$$\operatorname{spec}(H) = \{E_n\lt0\} \cup [0,\infty).$$

Bound states are square-normalized. Positive-energy states are generalized eigenfunctions with delta normalization and oscillatory asymptotics. A schematic completeness relation therefore contains both pieces:

$$I = \sum_{n\ell m} \lvert n\ell m\rangle \langle n\ell m\rvert + \sum_{\ell m} \int_0^\infty dk\, \lvert k\ell m\rangle \langle k\ell m\rvert.$$

Because the Coulomb tail falls only as $1/r$, continuum radial waves acquire a logarithmic asymptotic phase. They are not ordinary free waves plus a constant short-range phase shift. The repulsive Coulomb problem has continuum states but no hydrogenic bound ladder.

Continuum States of the Coulomb Problem: Overview places the threshold and generalized normalization beside the bound states. Detailed amplitudes and Rutherford scattering belong in Coulomb Scattering.

Reading route

1. Start with Central Potentials for rotational structure and quantum labels.
2. Derive the half-line equation in Radial Schrödinger Equation.
3. Learn to inspect fixed-$\ell$ sectors in Effective Radial Potential.
4. Fix endpoint and normalization conventions in Boundary Conditions for Radial Wavefunctions.
5. Establish signs and natural scales in Coulomb Potential.
6. Study the exact bound-state model in Hydrogen Atom.
7. Generalize the scales through Hydrogenic Ions.
8. Use Radial Wavefunctions as the formula, node, and expectation-value reference.
9. Interpret the complete spatial functions in Atomic Orbitals.
10. Separate rotational from hidden degeneracy in Degeneracy of the Hydrogen Atom.
11. Complete the spectrum with Continuum States of the Coulomb Problem: Overview.

Page map

Page	Canonical role
Central Potentials	Rotational invariance, separated states, quantum labels, and generic spectra
Radial Schrödinger Equation	$R$ and $u$ equations, normalization, regularity, and free radial motion
Effective Radial Potential	Centrifugal barrier, turning points, nodes, and qualitative radial spectra
Boundary Conditions for Radial Wavefunctions	Origin and infinity domains, finite-radius matching, and singular-potential cautions
Coulomb Potential	Charge signs, reduced mass, Bohr and Rydberg scales, and model limitations
Hydrogen Atom	Exact nonrelativistic bound spectrum, quantum numbers, and wavefunctions
Hydrogenic Ions	$Z$ and reduced-mass scaling for one-electron ions
Radial Wavefunctions	Laguerre structure, low-lying formulas, radial probabilities, nodes, and moments
Atomic Orbitals	One-electron orbital meaning, real and complex bases, and visualization
Degeneracy of the Hydrogen Atom	$n^2$ count, rotational versus Coulomb degeneracy, and splitting
Continuum States of the Coulomb Problem: Overview	Ionization threshold, Coulomb functions, delta normalization, and long-range phase

Common mistakes

• Treating “central” as a synonym for “Coulomb.”
• Assuming every central potential has hydrogen-like degeneracy between different $\ell$ sectors.
• Normalizing $R(r)$ with $dr$ or $u(r)$ with $r^2dr$.
• Treating the reduced radial equation as a full-line one-dimensional problem.
• Ignoring the origin when choosing the radial Hamiltonian domain.
• Calling the centrifugal term an independent interaction.
• Using a repulsive Coulomb sign while claiming hydrogenic bound states.
• Replacing the electron-nucleus reduced mass by $m_e$ without stating the approximation.
• Confusing the Bohr radius, the radial wavefunction maximum, and an atomic hard edge.
• Treating an orbital as a classical trajectory or its colored lobes as charged pieces.
• Counting $n^2$ degeneracy without distinguishing rotational and hidden-symmetry contributions.
• Treating the discrete hydrogen levels as the entire Coulomb spectrum.
• Applying short-range plane-wave asymptotics to the unscreened Coulomb continuum.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.

Exercises

1. A generic central-potential bound state is labeled $(n_r,\ell,m)$. State the degeneracy guaranteed by rotations and explain why changing $\ell$ usually changes the energy.

Solution

For fixed $n_r$ and $\ell$, rotational invariance guarantees equal energy for

$$m=-\ell,-\ell+1,\ldots,\ell,$$

so the multiplet has degeneracy $2\ell+1$. The radial equation contains

$$\frac{\hbar^2\ell(\ell+1)}{2\mu r^2}.$$

Changing $\ell$ changes this centrifugal term and hence changes the radial spectral problem. Degeneracy across different $\ell$ values therefore requires additional structure, such as the hidden symmetry of the Coulomb potential.

2. Compare two hydrogenic ions with the same quantum numbers but parameters $(\mu_1,Z_1)$ and $(\mu_2,Z_2)$. Find the ratios of their characteristic lengths and binding-energy magnitudes.

Solution

The Coulomb length and energy scales obey

$$a_Z\propto\frac{1}{\mu Z}, \qquad E_Z\propto\mu Z^2.$$

Therefore

$$\frac{a_2}{a_1} = \frac{\mu_1Z_1}{\mu_2Z_2},$$

and

$$\frac{\lvert E_2\rvert}{\lvert E_1\rvert} = \frac{\mu_2Z_2^2}{\mu_1Z_1^2}.$$

These ratios hold at fixed dimensionless state labels within the nonrelativistic point-Coulomb model.

3. For the hydrogenic shell $n=4$, list the allowed $\ell$ values, their radial node counts, and the total spatial degeneracy.

Solution

The allowed values are

$$\ell=0,1,2,3.$$

Using $n_r=n-\ell-1$ gives

$$\begin{array}{c|c|c} \ell & n_r & 2\ell+1\\ \hline 0 & 3 & 1\\ 1 & 2 & 3\\ 2 & 1 & 5\\ 3 & 0 & 7 \end{array}$$

The total spatial degeneracy is

$$1+3+5+7=16=4^2.$$

The equality among different $\ell$ sectors is the special Coulomb degeneracy; the entries within each $m$ multiplet are rotationally degenerate.

4. Why must a spectral expansion for the attractive Coulomb Hamiltonian include both a discrete sum and a continuum integral? State how the two classes of eigenfunctions are normalized.

Solution

The attractive Coulomb Hamiltonian has negative-energy bound states and positive-energy ionized states. The bound states are square-integrable and use ordinary normalization,

$$\langle n\ell m\vert n'\ell' m'\rangle = \delta_{nn'}\delta_{\ell\ell'}\delta_{mm'}.$$

The positive-energy states oscillate at infinity and are not square-integrable. They are generalized eigenfunctions normalized to a delta function, for example

$$\langle k\ell m\vert k'\ell' m'\rangle = \delta(k-k') \delta_{\ell\ell'} \delta_{mm'}.$$

A complete expansion must therefore sum over the discrete bound ladder and integrate over the continuum. Keeping only the hydrogenic orbitals omits ionized states and cannot represent arbitrary states in the full Coulomb Hilbert space.